Composition of Processes and Related Partial Differential Equations

Mirko D’Ovidio () and Enzo Orsingher ()
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Mirko D’Ovidio: “Sapienza” University of Rome
Enzo Orsingher: “Sapienza” University of Rome

Journal of Theoretical Probability, 2011, vol. 24, issue 2, 342-375

Abstract: Abstract In this paper various types of compositions involving independent fractional Brownian motions $B^{j}_{H_{j}}(t)$ , t>0, j=1,2, are examined. The partial differential equations governing the distributions of $I_{F}(t)=B^{1}_{H_{1}}(|B^{2}_{H_{2}}(t)|)$ , t>0, and $J_{F}(t)=B^{1}_{H_{1}}(|B^{2}_{H_{2}}(t)|^{1/H_{1}})$ , t>0, are derived by various methods and compared with those existing in the literature and with those related to $B^{1}(|B^{2}_{H_{2}}(t)|)$ , t>0. The process of iterated Brownian motion $I^{n}_{F}(t)$ , t>0, is examined in detail and its moments are calculated. Furthermore, for $J^{n-1}_{F}(t)=B^{1}_{H}(|B^{2}_{H}(\ldots|B^{n}_{H}(t)|^{1/H}\ldots )|^{1/H})$ , t>0 the following factorization is proved: $J^{n-1}_{F}(t)=\prod_{j=1}^{n}B^{j}_{\frac{H}{n}}(t)$ , t>0. A series of compositions involving Cauchy processes and fractional Brownian motions are also studied and the corresponding non-homogeneous wave equations are derived.

Keywords: Fractional Brownian motions; Cauchy processes; Modified Bessel functions; Iterated fractional Brownian motion; Mellin transforms; Fractional equations; 60J65; 60J60; 26A33 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-010-0284-9

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